# Analysis Seminar

#### Asymptotic Properties of Ground States of Scalar Field Equations with Vanishing Parameter

**Speaker:**
Cyrill Muratov

**Location:**
Warren Weaver Hall 1302

**Date:**
Thursday, April 5, 2012, 11 a.m.

**Synopsis:**

We study the leading order behavior of positive solutions of the equation $$-\Delta u + \varepsilon u - |u|^{p-2}u + |u|^{q-2}u=0, \qquad x \in \mathbb{R}^N$$ where \(N \ge 3\), \(q > p > 2\) and when \( \varepsilon > 0\) is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of \(p\), \(q\) and \(N\). The behavior of solutions depends sensitively on whether \(p\) is less, equal or bigger than the critical Sobolev exponent \(p^\ast=\frac{2N}{N-2}\). For \(p < p^\ast\) the solution asymptotically coincides with the solution of the equation in which the last term is absent. For \(p > p^\ast\) the solution asymptotically coincides with the solution of the equation with \(\varepsilon = 0\). In the most delicate case \(p = p^\ast\) the asymptotic behavior of the solutions is given by a particular solution of the critical Emden-Fowler equation, whose choice depends on \(\varepsilon\) in a nontrivial way.